Friday, August 11, 2006

Klein 2-Geometry IV

Can we sustain our momentum for the categorification of the Erlangen Program into its fourth month? At least now it is clear that what we need is a good account of how to quotient a 2-group by one its sub-2-groups. I've been messing around a little with some baby 2-groups and think I see how they work. I now think that the categorified Euclidean geometry that cropped up early on, i.e., the one that spoke of weak points and weak lines, arises from a discrete categorification of the Euclidean group. This has Euclidean transformations as 1-morphisms, and only trivial 2-morphisms. We may expect the geometry from more general 2-groups to look quite different.

Update: Things are hotting up. For the first time in my life (to my face at least) I've been called 'evil'. What can be achieved before the hiatus of a sojourn in France?

60 Comments:

From Ars Mathematica we read that John Iskra is writing a thoroughly category theoretic treatment of algebra - (Really) Modern Algebra. The title presumably refers to van der Waerden's Modern Algebra. Chapter 5 deals with groups, and might be a good basis to develop a systematic 2-group theory. I am a little surprised that more hasn't been done, although it could be buried in some text on bicategories.

Are there equivalents of the 3 isomorphism theorems? What is the proper definition of an abelian/soluble/nilpotent/simple 2-group? Is there a Jordan-Holder type theorem? Can 2-groups be defined in terms of generators and relations? The questions are endless.

Hi! Sorry to disappear on you: I've been hunkering down to get some work done. I just finished editing the galley proofs of my paper Quantum Quandaries, which will appear in Structural Foundations of Quantum Gravity by Steven French, Dean Rickles and Juha Saatsi. And, I finished polishing up the lecture notes taken by Mike Shulman, based on my talks on n-categories and cohomology at Chicago.

In my spare time, I've been working out a theory of "projective 2-geometry", based on the theory of 2-vector spaces that Alissa Crans and I came up with. I should tell you about it, because it seems like a natural example of this Klein 2-geometry business. It seems to work fine, but I still can't tell how interesting it is!

Please tell me sometime why you think the bits of Euclidean 2-geometry we came up with are associated to the discrete 2-group on the Euclidean group. I thought about it for a while and I think you're right, but I'd like to hear your reasoning.

You raise lots of questions:

Are there equivalents of the 3 isomorphism theorems? What is the proper definition of an abelian/soluble/nilpotent/simple 2-group? Is there a Jordan-Holder type theorem? Can 2-groups be defined in terms of generators and relations? The questions are endless.

I think a lot of these could be answered with existing knowledge about 2-categories and bicategories, withooo much struggle. But, for a lot of them, it's probably better to wait until

1) we generalize to infinity-groups

or

2) we have specific examples that make us need the answers to these questions.

For example, my work with Alissa on Lie 2-algebras was justified (in my opinion) by our discovery of a canonical 1-parameter family of Lie 2-algebras deforming any simple Lie algebra. For more general questions it would probably have been better to move right on to Lie infinity-algebras and think of them as Stasheff's L-infinity algebras, since he already has a substantIal technology developed.

You may think it's odd to put off certain questions until we answer them for infinity-groups, but remember, most people call infinity-groups topological groups or loop spaces, so they're already well studied; the only "new" thing is to study questions where everything is done "up to homotopy" - and even this isn't very new, since topologists have been doing that for loop spaces since the late 1960s. From this viewpoint, 2-groups are topological groups with only pi_0 and pi_1 nontrivial.

I'll answer a couple of easy questions, though!

Abelian 2-groups: looking at the periodic table of n-categories, one sees that one has 3 levels of abelianness in this column: 2-groups, braided 2-groups and symmetric 2-groups. All of them can be classified (in principle) using group cohomology, as I explain in my paper. The first people to do this explicitly were Joyal and Street, in their unpublished paper on braided monoidal categories, which is now available online, thank god!

Generators and relations: yes, 2-groups and indeed every sort of "purely algebraic" structure can be defined by generators and relations; this is part of the enormous generalization of universal algebra initiated by Lawvere.

I've been working out a theory of "projective 2-geometry", based on the theory of 2-vector spaces that Alissa Crans and I came up with.

It would be great to hear about this, and perhaps why projective geometry works here, and what would be different from other versions of 2-vector spaces, such as the one in Elgueta's paper - Generalized 2-vector spaces and general linear 2-groups, which discusses a GL(Vect[C]).

As for why I think our original categorified Euclidean geometry is based on the discrete categorification of the Euclidean group, I think it helps to consider what a quotient of 2-groups should look like.

Let's denote the 2-group we have been considering as G-H. Speaking loosely, this has G objects and G x H arrows. Consider a sub-2-group M-N. The M acts on G to give G/M objects (really G//M). Quotienting the arrows gives (G x H)/(M x N), so very loosely, H/N arrows emerging from each object (component).

Remember we called the 2d Euclidean group E, the stabilizer of a point P, and the translations T. The quotient E-1/P-1 has T or R^2 objects (each really a copy of P), and a single arrow out of any of these points, i.e., the identity arrow. That seems to be the plane of weak points we were considering. I think weak lines only go through weak points whose internal dials are pointing in their direction (something like a gauge theory with no turning of pointer along lines). There might be a richer geometry where these pointers can spin along a line.

If M-N is a normal sub-2-group, the quotient ought to be a 2-group too. We should expect that P-T is the quotient E-T/T-1. Your Poincare 2-group is a similar quotient.

As for why I think our original categorified Euclidean geometry is based on the discrete categorification of the Euclidean group, I think it helps to consider what a quotient of 2-groups should look like.

We certainly need to understand that if we're going to get anywhere with categorified Klein geometry.

Let's denote the 2-group we have been considering as G-H.

Yup. "Experts" write this 2-group as H->G, since since any strict 2-group gives a crossed module with a target map

t: H -> G.

There's also an action of G on H, and that's the only structure that the notation H->G conceals.

Yes; really the group of arrows is the semidirect product of G and H, since G acts on H - but apart from that you're not being "loose" at all here.

Consider a sub-2-group M

Let me make this a bit more precise, since I'm worried about cutting too many corners here. If we don't understand quotients of 2-groups, we can't do Klein 2-geometry.

Let's first suppose we have the 2-group MC//(Mshould be. If we then restrict to the case where C is a 2-group, we'll be sure to get the right answer.

(After all, one level down, the quotient of a group by a subgroup, viewed as a mere set, is nothing but a special case of the quotient of a set by a group.)

For the moment, just to spare both of us some pain, I'll only consider the case where Mstrictly on C: in other words, the usual laws for an action

c1 = 1

(cm)m' = c(mm')

hold as equations instead of up to isomorphism. This will be the case when we study a subgroup of a strict 2-group acting on this group by right multiplication - and we've only been talking about strict 2-groups lately.

So, how do we form the quotient

C//(MWe must use a weak quotient or we'll get in trouble somewhere down the line. So, we start with C and instead of quotienting by making some objects equal, we do so by throwing in new isomorphisms, which satisfy some equations called "coherence laws".

For each object c of C and m in M, we throw in an isomorphism

c -(c,m)-> cm

We impose equations saying these are natural in c and m. In other words, we make certain obvious squares commute whenever we have a morphism

c -f-> c'

or a morphism

m -f-> m'

(Test to see if you follow: which square can actually be drawn as a commutative triangle?)

We also impose some coherence laws related to multiplication and units in our 2-group. If we have m and m' in M, we can form

c -(c,m)-> cm -(c,m')-> (cm)m'

and also

c -(c,mm') -> c(mm')

These should be equal, or we'll be throwing in many different isomorphisms between things we're trying to "identify", instead of just one.

Similarly, we want

c -(c,1) -> c1 = c

to be the identity morphism.

(Test: show this is actually redundant.)

I think that's all. Quotienting by a weak 2-group action is not much harder; the only thing we need to remember is that instead of having equations (cm)m' = c(mm') and c1 = 1, we have isomorphisms, so we need to include those isomorphisms in the last two coherence laws I mentioned.

In my next post - I think I'll pause here so readers don't go insane with boredom - I'll see what we get when we mod out the "discrete Euclidean 2-group" by the "discrete rotation sub-2-group".

E(n)Here E is the Euclidean group in n dimensions, and we're making it into the discrete Euclidean 2-group ESimilarly, O(n) is the subgroup of the Euclidean group that stabilizes a point - better known as the rotation-reflection group - and O(n)discrete rotation-reflection 2-group.

Lastly, the "weak quotient" operation // was described in my previous post. (Nobody who writes in blogs talks like this!)

Putting them together, we see first of all that

E(n)has E(n) as its objects. What about the morphisms? We start out with just identity morphisms in E(n)e -(e,o)-> eo

for any e in E(n) and any o in O(n). The naturality squares I mentioned last time are all trivial, so the only coherence law we need is that

e -(e,o)-> eo -(eo,o')-> eoo'

equals

e -(e,oo')-> eoo'

What do we get?

This is just the weak quotient

E(n)//O(n) !!!

So, David was right.

We can draw a couple of morals from this. One is that whenever we have a group G and a subgroup H, we have

This stuff is important because it shows how Klein 2-geometry - the study of geometries with 2-groups rather than groups of symmetries - is related to our "automatic categorification" process that turns a homogeneous space G/H into a category G//H. We now see this category is indeed an example of a Klein 2-geometry, with the discrete 2-group Disc(G) as its symmetries.

So - good idea, David!

I think now we're ready for some Klein 2-geometries where the symmetry 2-group is a bit more interesting - not just a group in wolves' clothing. So, I should tell you about projective 2-geometry.

Okay, here's a taste of projective 2-geometry. I won't go into much detail now, since I've written too much on your blog already today. I'll just sketch a bit of what I did while sitting around at various Starbuckses - annoying plural, that! - around Shanghai.

(I only go to Starbuckses - Starbucki??? - when Lisa is shopping in tourist areas. That's about the only time I see white folks these days. For just 8 times the price of a bottle of ice tea at a typical Chinese store, you can get a steaming hot cup of coffee at Starbucks, perfect for a Shanghai summer. But, the caffeine blast is enough to make me dream up all sorts of crazy cool math, so the sweat is worth it.)

Before we categorify projective spaces, we should probably try to categorify vector spaces. I know three main ways to do this (and lots of small variations):

1) Kapranov-Voevodsky 2-vector spaces. These are categories equivalent to Vect^n, where Vect is the category of finite-dimensional vector spaces. They're very useful in topological quantum field theory, but they have no "negatives" of objects, much less the ability to multiply an object by an arbitrary complex number, so they're not an easy context for generalizing most of ordinary linear algebra.

2) Baez-Crans 2-vector spaces. These are categories with a vector space of objects, a vector space of morphisms, and with all the usual category operations (e.g. composition) being linear. Here you can take an object and multiply it by -1.73 + 42 i. There's a nice theory of categorified Lie algebras in this context, called "Lie 2-algebras", and these have corresponding "Lie 2-groups", etc..

3) Elgueta 2-vector spaces. These are free additive complex-linear categories on ordinary categories. In other words, to form one, we start with a category and formally throw in direct sums of objects and complex linear combinations of morphisms.One great thing about these is that you can form the "2-group 2-algebra" of a 2-group, much like the "group algebra" of a group.

Anyway, for my purposes I want to use 2) - not just because I helped invent this kind of 2-vector space, but because these are the most closely linked to Lie groups and other ideas from differential geometry. We're doing "Klein 2-geometry", and I'd like it to be related to geometry that we know and love!

Given a (complex) vector space V, the group C* of invertible complex numbers acts on V by scalar multiplication, and we can form the projective space PV like this:

PV = (V - {0})/C*

Given a 2-vector spaces V, the discrete 2-group Disc(C*) acts strictly on V by scalar multiplication, and we can form the projective 2-space PV like this:

PV = (V - {0})//Disc(C*)

Here {0} means something funny: it's the connected component of the object 0 in V. You can't just rip out an object from a category without pulling out all the morphisms from and to it! And, it would be insane not to pull out all isomorphic objects, too. All morphisms in a 2-vector space are isomorphisms, so when we rip out 0, all objects isomorphic to it, and all isomorphisms between them, we are removing the connected component of 0: the category consisting of all objects with morphisms from or to 0, and all morphisms between them.

I can describe what PV looks like very explicitly.

Up to equivalence, 2-vector spaces are classified by two natural numbers - the "first and second Betti numbers". (In general, n-vector spaces are classified by n numbers. When n = 1 this one number is called the "dimension".) It would be fun to see even more explicitly what PV looks like as a function of these two Betti numbers!

When the second Betti number is zero, our 2-vector space V is secretly just a vector space, and our projective 2-space PV is just an ordinary projective space.

All this is very nice: an orderly setup subsuming ordinary projective geometry as a special case.

I also suspect that these projective 2-spaces are "smooth 2-spaces" in the sense of Toby Bartels. These are the kind of category where you can do differential geometry: they're like manifolds in the world of categories.

For the purposes of Klein geometry, what matters more is that any 2-vector space V has a strict 2-group GL(V) acting on it, and this action commutes with scalar multiplication so we get GL(V) acting on PV. Presumably we get something called "PGL(V)" acting on PV, just as in ordinary projective geometry, but I haven't thought about that yet.

Also nice is that whenever we have a nontrivial "2-vector subspace" W in V, we get an inclusion

PW -> PV

These give "figures" like projective points, lines, etc. in our projective 2-space. But now the types of figures are indexed not just by dimension, but by two Betti numbers!

And, GL(V) will act on the set of figures of a given type.

So, this seems worth looking into a bit. If you ask me questions about stuff, that would give me an excuse to explain it in more detail.

For example, take C = Disc(K), for K some field and Disc(K) the discrete category with elements of this field as its objects. Then

Mod_C

is the 2-category of Baez-Crans 2-vector spaces over K, i.e. of categories internal to Vect_K.

On the other hand, for C = Vect_K we get a type of 2-vector spaces that live in

Mod_Vect_K,

or, if we agree on writing

Vect_K = Mod_K

in

Mod_Mod_K

(which should remind us of closely related similar recursive structures, like for instance

Tor_Tor_U(1) ).

Mod_Mod_K is pretty large. A nice tractable sub-2-category of Mod_Mod_K is

BiMod(Vect_K),

the sub-2-category of algebras, bimodules and bimodle homomorphisms internal to Vect_K.

(In order to regard this as a sub-2-category of Mod_Vect_K we need to send every algebra to the category of modules of that algebra and every bimodule to the functor obtained by tensoring with this bimodule.)

Now, Kapranov-Voevodsky 2-vector spaces are again a tiny sub-2-category of BiMod(Vect_K), namely that where all algebras involved are of the form K^n, for natural numbers n.

Elgueata's 2-vector spaces still live in a sub-2-category of Mod_Mod_K, somewhere in between the full thing and KV-2-vector spaces (including theses as a special case).

Is this aimed at the youth with their reduced attention spans? Perhaps we could get some background music going to liven it up. Did you read Fred Caligeri's comment in his review of a book on modular forms:

"With today’s iPod generation more likely to study elliptic curves and modular forms before learning any class field theory, Shimura’s book by itself is no longer apposite as an introduction to modular forms."

Up to equivalence, 2-vector spaces are classified by two natural numbers - the "first and second Betti numbers".

So, the first measures the dimension of the space of the objects, and the second the dimension of the space of morphisms with source 0, as in the 2-Term picture? Then if n_1 <= n_2, 0 will be connected to every object, so V - {0} is empty.

Something still niggling me is what the result of a 2-group acting on a category should be, or more generally what an n-group acting on a (n - 1)-category should be.

In the case n = 1, i.e., a group acting on a set, we either get a set S/G, or better a category S//G. Now for the case n = 2, you construct for us a category, by adding arrows and enforcing some equations. But why should we not expect a 2-category here? Presumably the answer to my question for general n according to your construction is "an (n-1)-category", but then why do we prefer the weak quotient in the case n = 1?

Sure. I would just like to keep in mind at which point we _have_to_ invoke special assumptions. One of my points was that the mere definition of a projective 2-vector space does not need the assumption that we have specialized to C = Mod_Disc(K).

David wrote:

"Then if n_1 <= n_2, 0 will be connected to every object"

Let the Baez-Crans 2-vector space be given by the 2-term complex of vector spaces

V_1 --d--> V_0 .

Then the zero vector object 0 in V_0 is connected to every other object in V_0 if and only if d is onto.

So a necessary condition for the isomorphism class of 0 to be V_0 is that dim(V_1) >= dim(V_0). But this is not sufficient. The kernel of d might be all of V_1, for instance, in which case our 2-vector space is skeletal.

I guess it is like passing to the skeleton. We pass from V_0 to coker(d).

But I might be wrong. I realize I need to think about what precisely we want to mean by writing V-{0}.

David wrote:

"Something still niggling me is what the result of a 2-group acting on a category should be, or more generally what an n-group acting on a (n - 1)-category should be."

A couple of entries before I argued that the _systematic_ way to derive this is by categorifying the concept of action groupoids by more or less straightforward internalization.

What's wrong with that?

I am pretty interested in the answer to that, albeit possibly for different reason than you are. I know what it means for a 2-bundle with connection to be equivariant under the action of an ordinary group. Strangely enough, it is not obvious at all (to me at least) what it would mean for it to be equivariant under the action of an honest 2-group. Apart from the problem of figuring this out by itself, I have no real clue in which applications a 2-equivariant 2-gerbe would arise naturally.

Sorry, I keep going off that gerbe tangent, while you want to have a discussion on Klein's 2-program.

Maybe I should make more explicit how I think that action 2-groupoid should look like, and how that reproduces what John wrote above (as far as I can see).

I originally pointed out that the ordinary action groupoid of G acting on C has space of objects equation to C and space of morphisms equal to GxC, with the obvious source, target and composition morphisms.

This straightforwardly internalizes in Cat, where now G is a 2-group, and C is some category with a G-action on it.

The result is a _double_ groupoid. And it really is a double groupoind non-equivalent to a 2-groupoid (in general) simply because horizontal and vertical morphisms will live in different categories.

If we assume everything in sight to be strict, we can easily draw the 2-cells of this action double-groupoid.

The category of vertical morphism is C.

The category of horizontal morphisms is C_0//G_0, i.e. the ordinary action groupoid at the level of objects.

If

c1 | | f | | vc2

is a morphism in C and

g1 | | h | | vg2

a morphism in G, then a 2-cell of our action double groupoid looks like

As I said in the comment section here (I cannot link to a specific comment here, can I?) we find the quotient of a group H by a subgroup G by forming the action groupoid of G acting on H and checking if this groupoid has the structure of a 2-group. If so, we form this 2-group and find that it is equivalent to a discrete 2-group Disc(H/G), which identifies the quotient group H/G that we are looking for.

I am guessing that the same procedure should apply here. Let C be a 2-group being acted on by the 2-group G. We form the action double groupoid as I indicated above.

Under special conditions (which we would address as saying that G is a _normal_ sub-2-group of C) this double groupoid should have the structure of a 3-group!

In other words, there should then be a way to take two of the 2-cells that I have drawn last time and put them on top of each other (in the third dimension) to produce a new 2-cell, such that this operation is double functorial and has inverses.

If this is the case, we should find a 2-group such that the resulting 3-group is equivalent to that 2-group, regarded as a 3-group with only identity 3-morphisms. This 2-group is the quotient group we are after.

While conceptually straightforward, I'd need pencil and paper to work out a nontrivial example, if any. Right now I have none with me.

However, the trivial examples, where both G and C are discrete, are easily seen to reproduce the expected result.

Well, I don't think people read blogs expecting a single entry to be a long disquisition, especially on a technical matter like Klein 2-geometry. Part of the problem is this damned "skinny column" format, which makes everything twice as long as it would otherwise be.

John wrote:

Up to equivalence, 2-vector spaces are classified by two natural numbers - the "first and second Betti numbers".

David wrote:

So, the first measures the dimension of the space of the objects, and the second the dimension of the space of morphisms with source 0, as in the 2-Term picture?

Not quite - I'm really talking about Betti numbers: dimensions of the homology groups of a chain complex.

Let me explain.

We've already seen how a 2-group can be written as a crossed module: a group G, a group H, a homomorphism

t: H -> G

together with an action of G on H, satisfying two axioms: equivariance and the Peiffer identity.

Just as a vector space is a special sort of group, a (Baez-Crans) 2-vector space is a special sort of 2-group. Viewed as a crossed module, it's simply one where G and H are vector spaces, and the action of G on H is trivial.

In this particular case, equivariance and the Peiffer identity are automatic, so we can forget about them.

So, a 2-vector space boils down to a couple of vector spaces G and H and a linear map

t: H -> G

This otherwise known as an operator, or, if we wish to show off, a 2-term chain complex of vector spaces. The latter terminology would be ridiculous overkill, were it not a special case of a more general fact: n-vector spaces are secretly just n-term chain complexes of vector spaces!

Now, chain complexes have homology groups, and the dimensions of the nth homology group is called the nth Betti number of our chain complex. It's a nice fact that, just as finite-dimensional vector spaces are classified up to isomorphism by their dimension, chain complexes of finite-dimensional vector spaces are classified up to equivalence by their Betti numbers.

In particular, 2-vector spaces are classified up to equivalence by their Betti numbers. Concretely, for the 2-vector space

d: V_1 -> V_0

the 0th Betti number is

dim (coker d)

while the 1st Betti number is

dim (ker d)

Here the kernel of d, "ker d", consists of V_1 that get sent to 0 by d, while the the cokernel of d, "coker d", is V_0 modulo the image of d.

For example, look at this 2-vector space:

1: C -> C

where C is the complex numbers. The dimensions of V_0 and V_1 are both 1, but the 0th and 1st Betti numbers are both 0! So, this 2-vector space should be equivalent to the trivial 2-vector space,

1: {0} -> {0}

And, it's easy to see this if we think of 2-vector spaces as categories (just as we do for 2-groups). The category corresponding to

1: C -> C

has C's worth of objects, and a single morphism from any object to any other. So, this category is equivalent to the category whose only object is 0, and whose only morphism is the identity. And this little category corresponds to

1: {0} -> {0}

In short: adding extra objects, but also extra morphisms saying these objects are isomorphic, gives us a new "puffed-up" version of a 2-vector space which is equivalent to the one we started with... and its Betti numbers don't change.

David wrote:

Then if n_1 <= n_2, 0 will be connected to every object, so V - {0} is empty.

Well, your guess about the definition of these numbers was a bit wrong, but your intuition is right, so we can fix what you're saying here. The 0th Betti number of our 2-vector space is the dimension of the space of isomorphism classes of objects. (The 1st Better number is the dimension of the automorphism group of any object!) So, if the 0th Betti number is zero, all objects are isomorphic to 0, and what I'm calling

V - {0}

will be empty.

This generalizes the fact that if V is a plain old vector space,

V - {0}

will be empty if its dimension is zero. Vector spaces give special 2-vector spaces, and their "dimension" then becomes the "0th Betti number". Nice, huh?

Moral: the projective 2-space PV is empty if the 0th Betti number of V vanishes.

Hi, Urs! Believe it or not, I'm writing a This Week's Finds about your ideas on the M-theory 3-group, in order to procrastinate from finishing my paper with Aaron before finishing my paper with you. But, I decided to procrastinate a bit on writing This Week's Finds, so I posted another article on David's blog here - and ran into you!

I'm procrastinating a lot these days, but no matter what I do, it always seems to involve higher categories, so I figure it's not so bad.

You write:

To me, a 2-vector space is

- a monoidal category C, playing the role of the ground ring/ground field

- a module category V over C, playing the role of a vector space over C .

These are great concepts, but it's probably good not to speak of "2-vector spaces" in such generality, since category theorists already have a perfect term for this concept: an action of a monoidal category on a category. This categorifies the usual concept of an "action" of a monoid on a set.

When our monoidal category is equipped with an addition as well as a multiplication, it's called a "rig category" - or often, a bit incorrectly, a "ring category". These like to act on categories equipped with their own addition, which we then could call modules of our rig category.

When our rig category acts a bit more like a field, perhaps then its modules deserve to be called 2-vector spaces. But, I don't know anyone who has axiomatized a concept of "field category".

ANYWAY, despite this terminological nitpicking, I really like your idea. At first I didn't believe that the Baez-Crans 2-vector spaces were precisely the same as modules of the rig category Disc(K) (where K is some field), but now it's looking like you're right. Did you check this carefully?

Hmm, maybe it's obvious that "working over Disc(K)" is the same as "working internal to Vect_K." Some kind of general abstract nonsense seems to be at work here.

The Kapranov-Voevodksy 2-vector spaces are indeed modules of Vect_K, but as you seem to point out, such modules need to be especially nice to qualify as Kapranov-Voevodky 2-vector spaces: they're basically the (finitely generate) free modules, of the form

(Vect_k)^n

Anyway, I hope you forgive me if I pay less attention to your "big picture" ideas than I should - I really want to keep zooming in on examples of Klein 2-geometries. I want to get to the point where I clearly see how something like projective geometry fits into a bigger categorified picture. So, at least on this blog, I'm going to sink my teeth into this problem with the tenacity of a bulldog, and not let go, no matter what tasty morsels you dangle in front of me.

And I'm now procrastinating here so as not to write my talk about history of maths for a workshop seeking a new epistemology of maths. And the case I was going to consider is the history you are delaying writing with Aaron because of your blogging here, in particular whether you are just writing a Royal-road-to-me account, and whether, if this is so, a sentiment of this accounts for this piece of modesty:

"As we approach the present we discuss the work of less famous scientists, stopping shortly after we mention the authors of the current paper, when it becomes impossible to sink any lower."

But back to some proper procrastinating, might it be that although 2-vector spaces are classified up to equivalence by 2 natural numbers, that it makes a difference to their 2-vector subspaces? E.g,

1: C -> C

and

1: {0} -> {0}

are equivalent, but the former has a nontrivial sub 2-space.

What would the passage from projective to affine geometry with your 2-vector spaces look like? We might try the (3,0) 2-vector space, form {V - {0}}//R*, and quotient the 2-group of projective transformations by those which preserve (0,0,1).

"Believe it or not, I'm writing a This Week's Finds about your ideas on the M-theory 3-group,"

Cool. It's a topic that people secretly know a lot about, without knowning that they know it - since they don't know the higher category theoretic picture that it fits in. So it certainly is a topic that can benefit from a TWF.

One big question that I long to figure out the answer to is:

Do we need "curvators" to properly understand (super)gravity as a theory of an n-connection, or does it suffice to enlarge the structure n-algebra by auxiliary fields and let the fake flatness conditions take care of everything, along the lines that I pointed out here.

I am hoping it is the latter, but I don't know yet.

John wrote:

"[...] before finishing my paper with you [...]"

For a long time I was worried that I would never be able to reformulate the proofs in a way that you would consider "finished". But now I know what we need.

My vacation ends today. I will start typing the new, better way to formulate all this starting tomorrow.

John wrote:

"At first I didn't believe that the Baez-Crans 2-vector spaces were precisely the same as modules of the rig category Disc(K) (where K is some field), but now it's looking like you're right. Did you check this carefully?"

I first saw this stated in one of Elgueta's papers, unless I dreamed it. I did check it mentally, without ever trying to write things down cleanly. So, no, I did not check it real carefully.

But it should essentially be trivial. Being a Disk(K)-module category makes both object as well as morphisms K-modules, hence K-vector spaces. Composition must respect the Disc(K)-action by functoriality, hence must be K-linear. Similar arguments make source and target maps K-linear. I think.

John wrote:

"Anyway, I hope you forgive me if I pay less attention to your "big picture" ideas than I should - "

Sure. You two go ahead and think about 2-Klein. I will now and then check which of your constructions require assuming restriction to Mod_Disk(K), and which don't.

In the end, somebody should sit down and try to see how the big theorems of linear algebra, i.e. the spectral theorem mostly, lift to Mod_C. I made some comments on that here. There are several indications that this is very important for 2D QFT.

"such modules need to be especially nice to qualify as Kapranov-Voevodky 2-vector spaces:"

Yes, they need to be categories of K^n-modules, instead of modules for arbitrary algebras, i.e. they live in BiMod_K^n.

I think what we should really be looking at is all of BiMod (regarded as a sub-2-category of Mod_Vect they way I indicated).

Why? For instance because every strict 2-group H-->G together with any rep of H has a canonical faithful 2-rep (->) in BiMod induced by that rep of H, which, for H-->G the string 2-group, essentially reproduces the Stolz-Teichner rep.

Similarly, the corresponding 2-rep of shifted U(1) coming from the canonical 1D rep of U(1) is the right one to find abelian gerbes as associated line 2-bundles. I am in the process of writing that out.

Please pardon the somewhat "disciplinary" tone of the following entry. I'm afraid David is being a bit... well, evil.

But back to some proper procrastinating, might it be that although 2-vector spaces are classified up to equivalence by 2 natural numbers, that it makes a difference to their 2-vector subspaces? E.g,

1: C -> C

and

1: {0} -> {0}

are equivalent, but the former has a nontrivial sub 2-space.

Right. But, we've run into the same puzzle before, back in July, when we noticed that the 2-group TRIV(G) was equivalent to the trivial 2-group but had nontrivial sub-2-groups. I wrote:

The really interesting puzzle is how a boring 2-group can seem interesting: for example, how a weakly trivial 2-group can have nontrivial sub-2-groups.

I thought I answered this puzzle - but now it looks like I never got around to it! Whoops!

The answer is that the naive concept of "subcategory" is evil. The same holds for "sub-2-group", "sub-2-vector space" and so on.

What do I mean by the naive concept of subcategory? I mean the one where we say C is a subcategory of D if there is a functor

C -> D

that's one-to-one on objects and on morphisms.

This is clearly evil, because "one-to-one on objects" is defined in terms of equations between objects, and equations between objects are always evil. You should always use specified isomorphisms instead.

But what do I mean by evil? This is a technical term here: it means "not invariant under equivalence of categories".

The point is that we want all true statements to remain true whenever we replace all the categories involved by equivalent categories. Life runs smoothly when this holds. It holds automatically if all our concepts are invariant under equivalence - that is, non-evil. But if we accidentally introduce an "evil" concept into our repertoire, we have to be incredibly careful. If we're not, various puzzles and "paradoxes" will emerge, in which equivalent categories start acting differently!

And, that's just what keeps happening when you (and perhaps even I) fling around words like "subcategory", "sub-2-group" and "sub-2-vector spaces", defined in a naive way.

The naive concept of "subcategory" is manifestly evil, since if C is a naive subcategory of D:

C -> D

and we have an equivalence

D -> D'

the composite

C -> D'

does not make C into a naive subcategory of D'.

So: instead of using naive, evil notions, we should use more sophisticated good ones. What are some good notions that we can use to talk about a functor

C -> D?

I can think of three. How can we use them to say that C is a subcategory of D, but in a less naive way?

A separate problem, by the way, is that you seem to be counting 2-vector subspaces of a 2-vector space and taking that number seriously. Counting in this naive way is good for sets, but evil for categories... and surely there should be something like a category of 2-vector subspaces of a given 2-vector space. I'm not sure how to define it - but surely that's what we want for Klein 2-geometry!

Thanks for pointing out my sins. I'll go flagellate myself with a category of nine isomorphism classes of tail.

What are some good notions that we can use to talk about a functor

C -> D?

I can think of three.

Snaffling a piece of your presence elsewhere in the blogosphere, here's one:

Makkai’s concept of an “anafunctor” F: C -> D, which assigns to each object in C not a specific object of D but only the universal property of an object in D.

I should think those Chicago notes contain more, perhaps in the appendix.

As for what about a functor wants to make you say it's a 'subcategory', there must be some construction from 2-category theory which is equivalent to 'monic' in 1-category theory. Something along the lines of F: C-->D is a subcategory if for any two functors G,H: B-->C with FoG naturally equivalent to FoH, then G is nat. equiv to H.

Thanks for pointing out my sins. I'll go flagellate myself with a category of nine isomorphism classes of tail.

It's probably better if you just quit being evil. For good. In real life this is hard, but in category theory it doesn't take being a saint: you just need to build a little mechanism into your conscience that automatically avoids "equations between objects".

This mechanism will make little warning bells chime before you speak of a "subcategory" in the naive sense as one whose inclusion functor

C -> D

is (among other things) one-to-one on objects. Since this concept involves equality between objects, it's almost sure to be evil.

Of course, you also need to have a built-in mechanism that takes the concept "sub-thingie" and automatically translates it into "monomorphism between thingies". No naive "subsets" here, please! Such concepts should be expressed in terms of arrows.

So, when you learn to shun your evil ways, before you say "subcategory" your conscience will cry "Wait! I mean something like monomorphism between categories!" And then it'll cry "Wait! How can I formulate this notion without mentioning equations between objects?"

Okay, that concludes today's sermon.

Next, I was trying to nudge you towards some answers to these questions of conscience....

What are some good notions that we can use to talk about a functor

C -> D?

I can think of three.

David replied:

Snaffling a piece of your presence elsewhere in the blogosphere, here's one:

Makkai’s concept of an “anafunctor” F: C -> D, which assigns to each object in C not a specific object of D but only the universal property of an object in D.

Anafunctors? We're talking about non-evil properties of functors right now, not replacements for functors. The three properties I was hinting at are well known to you:

faithful,full,essentially surjective.

These are all fundamentally "surjectivity" properties - I think we've talked about that before. But, "faithfulness" acts like an "injectivity" property, because while it really means (something like) "surjective on equations between morphisms", this turns out to mean (something like) "injective on morphisms".

And that's good, because right now we're looking for injectivity properties, since we're trying to understand monomorphisms between categories in a non-evil way.

So, we know how to say

C -> D

is "injective on morphisms" in a good way - we say it's faithful.

The question is, how to say it's "injective on objects" in a good way.

In other words, what should "essentially injective" mean???

I should think those Chicago notes contain more, perhaps in the appendix.

However, this stuff gets a bit involved, and I'm not sure it even answers the key question at hand, namely:

What's a non-evil way to say a functor is something like "one-to-one on objects"?

As for what about a functor wants to make you say it's a 'subcategory', there must be some construction from 2-category theory which is equivalent to 'monic' in 1-category theory. Something along the lines of F: C-->D is a subcategory if for any two functors G,H: B-->C with FoG naturally isomorphic to FoH, then G is nat. isomorphic to H.

Nice! Yes, that's a non-evil concept, and it's a bit like some things Mike talks about in Section 5.5 - but not quite the same. It might be just what we want, but I can imagine a very direct lowbrow non-evil way to express the idea of "one-to-one on objects", too.

The question is, how to say it's "injective on objects" in a good way.

In other words, what should "essentially injective" mean???

I'm not sure if that works, since if we have a skeleton skel(C) of a category C, the desired `subcategory' A \to C needs to be a subcategory of skel(C), as a category is equivalent to any of its skeletons.

So we send each object in C to a chosen representative of its isomorphism class to get skel(C), and the composite A \to C \to skel(C) is certainly not injective on objects.

Street's old paper ``Two-dimensional sheaf theory''has as one of its aims a 2-categorical version of a regular category (strict, mind you) and he defines an arrow (e.g. a functor) in a 2-category to be chronic when it is fully faithful and injective on objects. In the bicategorical version of this, he drops the injective on objects. He also discusses so-called acute arrows which act like regular epimorphisms in a category (e.g. G \to G/H for a normal subgroup H). One could then possibly define normal 2-subgroup to be the kernel of such a morphism in the bi-/2-category of 2-groups. I think Vitale and others have considered what the kernel of a map between 2-groups is. Actually, now that I think about it, Vitale did a bit more that that in ``A Picard-Brauer exact sequence of categorical groups''.

Can we get categorical versions of the isomorphism theorems, with isomorphism replaced by equivalence? This is a very relevant point for the Klein part of this discussion (I apologise like the rest for diversions) - how `transitive' are subfigures? How do we compare G//H to

(G//K)//(H//K)

for K normal in H normal in G? Or has this been done while I wasn't watching?

Back to a old topic, I always wondered what an equivalence relation on a category was, given an equivalence relation on a set is a groupoid. Here of course we are interested in the 2-equivalence relation: ``Is in the same orbit of the given 2-group''. This sould be much easier to figure out than the general case.

Here's an example close to my heart: Take the one point topological space pt with a G-action (G a compact Lie group for a comfortable example). (Homotopy) quotient it. The map q: pt \to pt//G is onto. If G is abelian, we can take pt of course to be the trivial group and we can get a group structure on pt//G (in fact a 2-group structure!). What is the kernel of q?

Also, does a normal 2-subgroup have to be invariant under the ``conjugate'' action, or only up to equivalence?

One other point: when we get to Baez-Crans 2-vector spaces, what happened to the Postnikov information (=associator) one gets when classifying 2-groups?

I can imagine a very direct lowbrow non-evil way to express the idea of "one-to-one on objects", too.

Something like F: C-->D is one-to-one on objects whenever, for any pair of objects X,Y in C, if F(X) and F(Y) are isomorphic in D, then X and Y are isomorphic in C.

Might that be called 'full on isomorphisms'? Pause for quick Google search. Aha!

"...we call a functor U :A -->C pseudomonic if it is faithful and if, moreover,it is full on isomorphisms: the latter means that any invertible h : UA --> UA' in C is Ug for some (necessarily unique) g : A --> A' in A, which by an easy argument must itself be invertible." p. 230 of this.

I'm not sure if that works, since if we have a skeleton skel(C) of a category C, the desired `subcategory' A \to C needs to be a subcategory of skel(C), as a category is equivalent to any of its skeletons.

So we send each object in C to a chosen representative of its isomorphism class to get skel(C), and the composite A \to C \to skel(C) is certainly not injective on objects.

Isn't this precisely why John said essentially injective on objects? The pseudomonic construction seems to deal with your example.

Does it make life easier working with skel(2Vect), along the lines of:

We also need to make sure that the image of our monic functor is indeed a category. The concept of essential injectivity must be strong enough to ensure that.

For, in general, the image of a functor is not a category, because the image of morphisms may become composable while the pre-images were not.

I don't know if this is terribly relevant to the present discussion. But we were concerned with this issue when I visited Zoran Skoda and Igor Bakovic in Zagreb a while back, where we tried to understand coequalizers in Cat in order to understand associated 2-bundles.

I am aware of this paper which deals with the issue by passing, essentially, to the smallest subcategory which contains the image of a given functor.

As all maps are isos, we need a full and faithful functor, which must come from 2-vector spaces with 0th Betti number less than or equal to b_0, and 1st Betti number equal to b_1. If correct, isn't this wrong:

Also nice is that whenever we have a nontrivial "2-vector subspace" W in V, we get an inclusion

PW -> PV

These give "figures" like projective points, lines, etc. in our projective 2-space. But now the types of figures are indexed not just by dimension, but by two Betti numbers!

See, David? I knew that if you let me call you "evil", your blog would get some new readers. Blogs thrive on conflict; we're too nice to actually fight, so we have to invent technical terms that make it look like we're fighting.

If you search for where John Baez says,

"I'm afraid David is being a bit... well, evil"

you'll see he meant it in a technical sense:

"But what do I mean by evil? This is a technical term here: it means "not invariant under equivalence of categories"."

Yes, I'm afraid David is not invariant under equivalence of categories.

Seriously, the fascinating thing about categorification is that by deliberately excluding certain syntactically well-formed concepts (those involving equations between objects), one obtains a more interesting theory! At present, we mainly accomplish this by moral suasion - arguing that certain concepts are "good" and others "evil". I'm highlighting this in a jokey way by making "evil" into a precise technical term.

But, it may not be satisfactory in the long run to exclude certain concepts by moral pressure. The logician Michael Makkai has grabbed the bull by the horns and made infinity-categories into a new "foundation for mathematics" in which there are no equations! This system is called FOLDS, or first-order logic with dependent sorts, and David and I saw Makkai explain it at the IMA conference on n-categories a couple summers ago. In this system you just can't say anything evil. It may never catch on, but it's a truly bold conception.

I can imagine a very direct lowbrow non-evil way to express the idea of "one-to-one on objects", too.

David wrote:

Something like F: C-->D is one-to-one on objects whenever, for any pair of objects X,Y in C, if F(X) and F(Y) are isomorphic in D, then X and Y are isomorphic in C.

Right! That's one straightforward way to weaken the definition of "one-to-one on objects". Jim calls such a functor essentially injective.

David wrote:

Might that be called 'full on isomorphisms'? Pause for quick Google search. Aha!

Oh, interesting. But notice,"full on isomorphisms" does not mean the same thing as "essentially injective" - it's a stronger property:

"...we call a functor U :A -->C pseudomonic if it is faithful and if, moreover,it is full on isomorphisms: the latter means that any invertible h : UA --> UA' in C is Ug for some (necessarily unique) g : A --> A' in A, which by an easy argument must itself be invertible.

You see, he's not just saying "if UA and UA' are isomorphic, then A and A' are isomorphic". He's saying every isomorphism between UA and UA' is the image of an isomorphism between A and A'!

I don't feel very knowledgeable about this stuff. So, it's quite possible that "full on isomorphisms" is the notion we really want, not "essentially injective".

In general, you can do a lot more if you talk about specific isomorphisms rather than "isomorphicness". It's really helpful to get rid of the existential quantifier built into the latter concept. This suggests that "full on isomorphisms" is more useful than "essential injectivity".

So does the fact that someone already decided to define pseudomonic to mean faithful and full on isomorphisms!

So, I need to do some reading and thinking, but we might wind up using pseudomonics instead of naive, evil "subcategories".

Urs wrote:

We also need to make sure that the image of our monic functor is indeed a category.

The naive concept of image of a functor

F: C -> D

is evil, since it contains those objects that are equal to objects of the form F(c). An object isomorphic to one in the image might not be in the image.

There are various non-evil substitutes for the notion of image; Toby uses the "full image" in his notes on properties, structure and stuff, and this is a category - but it's not what we really want here.

... in general, the image of a functor is not a category, because the image of morphisms may become composable while the pre-images were not.

Right. I claim that annoying behavior like this is to be expected, since "image" is an evil concept. But in some cases, the image will be a category.

Doesn't the faithfulness condition handle this? If the images compose, then there is an arrow, not necessarily the composition, whose image is the composition.

It sounds to me like you're using fullness here, not faithfulness. The image of a full functor is a subcategory for the reason you said. Remember, a functor is full if any morphism between objects in the image is in the image.

I think we can see that the image of a pseudomonic doesn't suffer from the problem Urs mentions. Suppose

F: C -> D

is pseudomonic: that is, faithful and full on isomorphisms. Suppose

f: d -> d'

and

f': d' -> d''

are morphisms in the image of F. Why is their composite in the image of F?

We have

f = F(g)f' = F(g')

but as Urs points out, there's a problem! g and g' might not be composable:

g: c -> c'

g: c'' -> c'''

but we may not have c' = c''. We only know

F(c') = d' = F(c'')

But now we use the fact that F is full on isomorphisms! There's an isomorphism

1: F(c') -> F(c'')

so this must be the image of some morphism

h: c' -> c''

This h bridges the annoying gap between c' and c''. Now look at

ghg': c -> c'''

where I'm composing morphisms left-to-right. We have

F(ghg') = F(g)F(h)F(g') = f1f' = ff'

So, ff' is indeed in the image of F! Hurrah! QED!

Now let's do a little postmortem on this proof.

First, we only needed the fact that F was full on isomorphisms. Faithfulness was irrelevant.

Second, while "full on isomorphisms" gets the job done, I don't think the weaker "essential injectivity" would work. We need to take a specific isomorphism in D:

1: F(c') -> F(c'')

and find a morphism in C that maps to it. It would not suffice to say "since F(c') and F(c'') are isomorphic, c' and c'' must be isomorphic."

This is the kind of thing I meant when I said it's more powerful to work with specific isomorphisms than "isomorphicness", i.e. mere existence of isomorphisms. I hadn't known it would be important here, but it's a lesson one learns over and over - and we're learning it again here.

So, for better or worse:

Theorem: The image of a functor

F: C -> D

is a category if F is full on isomorphisms.

I'm sure this is already known by those wiser than I.

Note this theorem is stated in an evil manner, since it uses the concept of "image". However, the concept of "image" is not actually evil when F is full on isomorphisms, since any object isomorphic to one in the image is again in the image!

So, pseudomonics seem like the right substitute for naively defined "subcategories".

Or, there could be more than one good substitute. I'm actually inclined to believe this, due to some reflections on properties, structure, stuff and eka-stuff. But, pseudomonics at least seem like one good substitute.

Can we get categorical versions of the isomorphism theorems, with isomorphism replaced by equivalence?

How do we compare G//H to

(G//K)//(H//K)

for K normal in H normal in G? Or has this been done while I wasn't watching?

No, we haven't gotten this far yet. We're still struggling to understand the notion of "sub-2-group"! That's how we got into studying various notions of "subcategory": first an evil notion proposed by David, and now an improved one, namely "pseudomonic functor".

So, now we can define a sub-2-group of a 2-group G to be a 2-group H equipped with a pseudomonic homomorphism

i: H -> G

In case you're wondering, a 2-group homomorphism was defined in HDA5 to be just a weak monoidal functor between 2-groups.

As Urs and you pointed out, we could have short-circuited some of this work if we'd only wanted to understand "normal" sub-2-groups, since we could define them as kernels of homomorphisms. But, since we're doing Klein 2-geometry, we really need to understand quotients

G//H

where H is an arbitrary sub-2-group of G. These are the "2-spaces of geometrical figures" in Klein 2-geometry.

So, we're getting close to tackling questions like the one you mentioned, but we're not there yet.

One other point: when we get to Baez-Crans 2-vector spaces, what happened to the Postnikov information (=associator) one gets when classifying 2-groups?

Ah, good - a question I can answer. Short answer: it's trivial!

Long answer: 2-groups are secretly the same as connected pointed spaces with only pi_1 and pi_2 nontrivial - all higher homotopy groups vanishing. So, they're classified up to equivalence by:

pi_1: the group of isomorphism classes of objects.

pi_2: the group of automorphisms of the identity object.

an action of pi_1 on pi_2 by "conjugation".

an element of the 3rd cohomology group of pi_1 with coefficients in pi_2, coming from the associator.

(This is explained in HDA5 and, in a more hand-waving way, in my notes on n-categories and cohomology, in the section called "A Low-Dimensional Example".)

The last two bits of information - the action and the 3rd cohomology class - are called Postnikov data. What happens if we make them trivial - make the action trivial and the cohomology class vanish? Suppose just for good measure that we also make pi_1 abelian! What do we get?

Then we get a specially simple sort of 2-group, which is secretly just a 2-term chain complex of abelian groups!

2-vector spaces are the special case of this where our abelian groups are vector spaces.

All this generalizes massively! Chain complexes of abelian groups are secretly just connected pointed spaces with pi_1 abelian and all Postnikov invariants trivial. Such spaces are just products of Eilenberg-Mac Lane spaces:

What is the proper definition of an abelian/soluble/nilpotent/simple 2-group?

I've been meaning to say something about this for a long time. I already said that 2-groups come in 3 amounts of abelianness: 2-groups, braided 2-groups and symmetric 2-groups. In general, n-groups come in n+1 different flavors, increasingly abelian as we march down the periodic table.

But what about "simple" n-groups?

There really are no simple n-groups except for simple 1-groups - that is, ordinary simple groups. By definition, a "simple" gadget has no nontrivial quotients. We can always form a quotient of an n-group which is an (n-1)-group, by decategorifying it. This quotient is nontrivial except when n = 1.

This leads to the theory of Postnikov towers, explained in my lecture notes. Using this theory, we can describe general n-groups as built up from simple groups, "glued together" using group cohomology.

The most familiar case is how we build up arbitrary groups from simple groups via iterated extensions. Extensions are classified using (nonabelian) 2nd cohomology. To build up n-groups, we need higher cohomology classes, up to the (n+1)st cohomology. For 2-groups, a 3rd cohomology class describes the associator - I described this in my last post.

So, the classification of finite simple groups is the first step on the road to classifying all finite n-groups. This road may well be too long and twisty for humans to ever see the end of it. Even classifying all finite groups is beyond us! But, we can still get lots of useful information by trying to understand how general n-groups are built up out of simple groups.

I'm thinking of G,H as monoidal cats here, not one-object 2-cats.Since we classify 2-groups up to these homotopy groups (with some Postnikov data) how do we say `pseudomonic' in this language, or have we forgotten too much information. Well, since 2-groups are groupoids, all morphisms are isos, so pseudomonic is an overly large tool for the job.

In this system you just can't say anything evil. It may never catch on, but it's a truly bold conception.

Once we get it to work in math, all we'll need is to find a similar language for the rest of life. Trouble is the relation being saying and doing is somewhat different there.

After my last posts, and the realisation while watching a film that I'd made all kinds of mistakes, as John pointed out, I was going on happily to think about the automorphism 2-group of a 2-vector space (the film being a bit dull).

Now, I return and pseudomonics are flavour of the month. So how about the problem I raised:

What are the pseudomonics into

d:C^b_1 --> C^b_0, with d the zero map

As all maps are isos, we need a full and faithful functor, which must come from 2-vector spaces with 0th Betti number less than or equal to b_0, and 1st Betti number equal to b_1.?

In general, doesn't this make the range of sub-2-groups of a 2-group rather boring? For instance, that example from way back about symmetries of two triangles, we were looking at a 2-group with 72 objects and 72 x 36 morphisms, and thought we were considering a sub-2-group with just identity morphisms. According to the pseudomonic story it's not a sub-2-group. It wasn't even essentially injective on objects.

we could have short-circuited some of this work if we'd only wanted to understand "normal" sub-2-groups, since we could define them as kernels of homomorphisms.

Do we know that the inclusion of a kernel of a 2-group homomorphism is pseudomonic (or even injective on objects)? Is the definition of kernel fixed? Is there a notion of 'essential kernel'?

The really interesting puzzle is how a boring 2-group can seem interesting: for example, how a weakly trivial 2-group can have nontrivial sub-2-groups.

I thought I answered this puzzle - but now it looks like I never got around to it! Whoops!

Now you answer it by saying that evil has intruded. But in the original context of the question back in July, you pointed out:

When people tried to categorify the theory of K-bundles, without quite knowing at first what they were doing, they basically invented the theory of AUT(K)-2-bundles - but under a different guise: they called it the theory of "nonabelian K-gerbes".

and that for some K, AUT(K) is equivalent to KI knew one needed fortitude to pursue mathematical research, but now I'm feeling that need. 4 months in and we're still wondering what a sub-2-group is. Is there any glimmer of gold ahead, or will it be iron pyrites for the foreseeable future?

I knew one needed fortitude to pursue mathematical research, but now I'm feeling that need. 4 months in and we're still wondering what a sub-2-group is. Is there any glimmer of gold ahead, or will it be iron pyrites for the foreseeable future?

Welcome to mathematics! Remember how I wondered in July if we'd have the energy to stick with this stuff?

Well, I know I do have the energy - I'm really happy about how things are going now. If you get tired, don't feel bad. You can quit if you like, now: you helped me over the hump, and that's all I could ask.

Now we know what projective 2-geometry is! Even better, we've quit fiddling around and started doing some serious work - like figuring out what a sub-2-group really is. It's subtler than one would have guessed! Before, we were just sort of crossing our fingers and hoping everything would work more or less like the decategorified case. That's good for starters, but now we've moved beyond that stage. Lemmas are starting to accumulate; we're sifting through candidate definitions - "essentially injective" versus "full on isomorphisms", that kind of stuff. In short: it's not just dreaming anymore; it's becoming a real subject, with ties to homological algebra (chain complexes), homotopy theory (Postnikov towers), algebraic geometry (those projective 2-spaces) and more. In another year or two, something impressive might actually happen.

This is how it always goes. Perhaps you're think ugh, it's getting to be a technical mess when I'm think hey, there are some really interesting questions here to straighten out - it's not all trivial!

It's really late now; tomorrow I'll tell you some stuff about sub-2-groups.

Well it wasn't quite up there with Henry V's speech on the eve of Agincourt, but it'll do.

Well, the troops being rallied aren't about to get riddled with arrows, either.

Or maybe they are....

I'm no longer sure "pseudomonics" are the right substitute for "monics" when we go from groups to 2-groups. I spent several hours at that Starbucks, cranking out math in a caffeinated frenzy, so I'll try to briefly summarize what I found. But first, I want to emphasize some excellent points D. M. Roberts made.

For starters, if we think of a 2-group as a category, every morphism in it is an isomorphism. So, for maps between 2-groups, "full on isomorphisms" just means "full", and "pseudomonic" means "full and faithful". Fewer nuances to worry about - good!

Second, when we're trying to understand maps between 2-groups, we should focus on data that's invariant under equivalence. As I mentioned a while back, a 2-group is known up to equivalence if we know its pi_1, its pi_2, the action of pi_1 on pi_2, and a 3rd cohomology class. Similarly, a map between 2-groups

F: G -> H

is known up to equivalence if we know the induced maps

pi_1(F): pi_1(G) -> pi_1(H)

pi_2(F): pi_2(G) -> pi_2(H)

So, it should not at all be surprising that interesting properties of F can be phrased in this language. D. M. Roberts gave us a handy dictionary. Let me flesh it out a bit, and translate it into my favored numbering scheme:

F is essentially surjective iff pi_1(F) is surjective.

F is essentially injective iff pi_1(F) is injective.

F is full on automorphisms if pi_2(F) is surjective.

F is faithful iff pi_2(F) is injective.

And some secondary notions:

F is full iff pi_1(F) is injective and pi_2(F) is surjective.

F is an equivalence iff pi_1(F) and pi_2(F) are isomorphisms

The only one of these that seemed tricky to me was "full". Remember, F is full if given a morphism

f: F(x) -> F(y)

we have

f = F(g)

for some

g: x -> y

We say F is full on automorphisms if this holds in the special case where x = y and f is an automorphism.

It's just a matter of definition-chasing to see that if a functor F between groupoids is full, it's full on automorphisms and also essentially injective.

Conversely, a functor F between groupoids is full if it's full on automorphisms and also essentially injective!

After all, F is "essentially injective" iff whenever there's some isomorphism between F(x) and F(y), there's some isomorphism between x and y. When we combine this with "full on automorphisms", we see that for any specific isomorphism

f: F(x) -> F(y)

we can find an isomorphism

g: x -> y

mapping to it. Why? Well, once we know x and y are isomorphic, there's no real distinction between x and y, so "full on automorphisms of x" is really the same as "full on isomorphisms between x and y".

Now, what about sub-2-groups?

Well, it's tempting to define this concept in terms of the ones we've just listed. If we follow our gut instincts, we might say a 2-group homomorphism

F: G -> H

exhibits G as a sub-2-group of H if pi_1(F) and pi_2(F) are both injective. By Roberts' dictionary, this means F is essentially injective and faithful.

This is not the same as "pseudomonic", which in this context means "full and faithful" or essentially injective, faithful and full on automorphisms.

See the culprit? Being "full on automorphisms" is a kind of surjectivity, not a kind of injectivity.

It's too bad the comments here aren't dated, now that this is becoming a kind of communal research diary. Oh well: let it be noted that on Sunday, July 13th I thought about the following things in the Starbucks right next to the Nine Zigzag Bridge.

I was desperately trying to understand sub-2-groups. So, I thought: in Klein geometry, the conceptual meaning of "subgroup" is really "stabilizer of some point in a set on which a group acts".

So, let's take a 2-group G acting on a category X, and let's study the the stabilizer of some object x in X. Whatever this stabilizer is like, maybe this should become the definition of a sub-2-group!

(Or, maybe not - there are also stabilizers of things more complicated and interesting than a mere object. But never mind! - it's still an interesting exercise.)

Of course we need to define the stabilizer, say Stab(x). There's an obvious way to do this if you're careful not to be evil. I'll just sketch it.

The stabilizer Stab(x) is a 2-group with the following objects and morphisms. An object of Stab(x) is an object g of G together with an isomorphism

is a morphism f: g -> g' in G making the obvious triangle commute. Namely,

a: gx -> x

should equal the composite of

fx: gx -> g'x

and

a': g'x -> x.

It really looks much prettier as a triangle!

With some work one makes Stab(x) into a 2-group - I didn't check everything here, but I'm following the tao of mathematics so I'm sure everything works, even when G is a weak 2-group and its action on X is also weak - the general case. I also feel sure we get a 2-group homomorphism

i: Stab(x) -> G

By the philosophy I described, this "inclusion" of Stab(x) in G should be a great example of a sub-2-group!

But, even if the above went by in a blur, you'll surely note that objects of Stab(x) are objects of G equipped with extra structure!

That's a bit scary, since it means we're "losing something" when we map from Stab(x) to G - not what we'd naively expect when dealing with "the inclusion of a sub-2-group"!

But, what are we losing, exactly? The point is that

i: Stab(x) -> G

"forgets extra structure". There can be lots of different ways to make an object of G into an object of Stab(x) - lots of different ways for a guy in a 2-group to stabilize a guy in a category it's acting on! Or, there can be no way at all! Choosing such a way is choosing extra structure.

If one is familiar with the yoga of properties, structure and stuff, one knows that "forgetting extra structure" means our functor is faithful, but it can fail to be full or essentially surjective.

By D. M. Roberts' handy chart, as polished by yours truly:

i is faithful: pi_2(i) is injective. YES.

i is essentially surjective: pi_1(i) is surjective. MAYBE NOT.

i is full: pi_1(F) is injective and pi_2(F) is surjective. MAYBE NOT.

We should not be surprised that something about i fails to be surjective. We may be surprised if something about it fails to be injective - since it's the very role model of an "inclusion of a sub-2-group".

So, the only surprising thing on the list is that pi_1(i) may not be injective. In other word: i may not be "essentially injective"! Objects that are not isomorphic in the sub-2-group may become so in the 2-group!

But this is not really surprising if you think about it. When we go from a sub-2-group to a 2-group, we can throw in new morphisms, which can make objects isomorphic that weren't before.

Moral: the inclusion of a sub-2-group

i: H -> G

had damn well better be faithful (injective on pi_2), but it might not be essentially injective (injective on pi_1).

I learned something else about

i: Stab(x) -> G

too! Namely, it's a discrete fibration of groupoids. Any morphism in G lifts uniquely to Stab(x) once we've lifted its source.

The fact that

i: Stab(x) -> G

is a fibration is nice - but in a sense no big deal, since any functor can be "improved" to be a fibration. The fact that it's a discrete fibration - the uniqueness of the lifting - is more interesting.

Okay, that's basically it for Sunday's Starbucks session. I'm sure it's too much for almost everyone!

The free groupoid on a graph + free group on the set of vertices should give us something like a free strict 2-group. the homotopy quotient by some sub2-group should give us what we want. So what is a 2-group of relations? Not equalities, but isos between objects (we have them) and a relation on morphisms.

On quotienting, it's almost (trying not to be evil) like we make some objects equal, keep the maps between them as automorphisms and identify some morphisms as the same. That is very hand-waving and apologies for it. I feel like this operation is like in homotopy when one when one contracts a subcomplex (objects becoming equal) or throws cells in to fill up unwanted spheres - in this case we are in very low dimensions, so circles and disks (morphisms becoming equal). Or am I completely wrong?

I can't think immediately of some homotopy group explanation/long exact sequence where one gets this stuff from.

It's too bad the comments here aren't dated, now that this is becoming a kind of communal research diary. Oh well: let it be noted that on Sunday, July 13th I thought about the following things in the Starbucks right next to the Nine Zigzag Bridge.

But they are dated, and just as well as you got the month wrong! I hope you didn't doing any carving into the Nine Zigzag Bridge

Do we know that the inclusion of a kernel of a 2-group homomorphism is pseudomonic (or even injective on objects)? Is the definition of kernel fixed? Is there a notion of 'essential kernel'?

Perhaps this was too obvious, but here we go: is the kernel of a 2-group homomorphism F:G \to H the homotopy fibre (call it K) over the tensor unit I? (thinking of the beasties as monoidal cats - otherwise as the fibre over the identity of the unique object if they were bigroupoids).

That is, objects of K are objects g in G such that F(g) is isomorphic to I, and morphisms are(iso)morphisms f in G such that F(f) = Id_I.

I haven't time to check the properties of i:K \to G, but it is certainly a truly injective functor. If h is iso to g in G and g is in K then h is in K, so we get whole isomorpism classes at least. There should also be a universal property here so the kernel is defined only up to equivalence, but like when we talk about pullbacks, this seems to be a nice explicit description.

One other point, echoing the point that an n-vector space is a n-term chain complex of vector spaces, I seem to recall that a strict n-group should be a simplicial group with only n-terms of its Moore complex non-zero. I don't want to define the Moore complex here (it's nasty), but to say the least, it gives a chain complex of non-abelian groups (the construction is such that all images are normal). For a 2-group, the crossed module G_1 \to G_0 is that Moore complex.

had damn well better be faithful (injective on pi_2), but it might not be essentially injective (injective on pi_1).

Just to get this straight, let's return to my original sin, where evil first became apparent in Paradise.

I said:

might it be that although 2-vector spaces are classified up to equivalence by 2 natural numbers, that it makes a difference to their 2-vector subspaces? E.g,

1: C -> C

and

1: {0} -> {0}

are equivalent, but the former has a nontrivial sub 2-space.

I was referring to the 2-vector space i:{0} -> C, which appeared to be a subspace of the former but not the latter. Now, according to the current definition, as this subspace is clearly faithful into C -> C, it must be a subspace. So it had better also be a subspace of {0} -> {0}. And, Lo and Behold, it is!

Hmm, any discrete 2-vector space is a sub 2-vector space of the trivial one! The only criterion to satisfy for being a sub 2-vector space of a 2-vector space with Betti numbers (b_0, b_1) is to have first Betti number no larger than b_1.

Hmmmm, but then can't we have (6, 0) as subspace of (3,0) and vice versa, without them being equivalent?

Either I'm making some obvious blunder, or the world of 2-groups is quite strange.

Yes, it's great. This is the sort of web discussion that I like a lot.

Since you like it, too, and if we all feel that the software running this is not particularly inspiring, maybe we should think about starting -- a new group blog.

Some blog with advanced software, hosted by maybe John, David, myself, possibly others - filled with lots of the sort of discussion that we apparently all enjoy.

If you don't like this idea, never mind. If it sounds at least interesting, then the most immediate thing I could do about this is to ask Jacques Distler if he would be willing to set up and administrate such a blog for us.

Jacques originally offered to set up the string coffee table, which runs the same software as his personal blog, when he heard me and others talk about the desire for an online place for disucssion of string theory.

You are all probably aware how the story continued, the outcome being the somewhat strange situation we have now.

So, I could ask Jacques if he would maybe abandon the SCT in favor of a true group blog - research diary style.

That's just the most immediate option I can think of. If you like the idea of a group blog, but don't want to depend on a third party administrating it (although to me this is a feature, not a bug) we could of course also try to set up the software ourselves.

Ob(K) = g in G and j_g an arrow of H such that h:F(g) --> I_H, for I_H the tensor unit in H

Arr(K) = a an arrow of G, a:g1 --> g2 such that F(a) form a commutative triangle with j_g1 and j_g2

Here (g1,j_g1) --a--> (g2,j_g2) is the arrow in K.

The inclusion map to G just forgets the extra structure given by the isomorphism j_g

This looks awfully similar to John's definition of the stabilizer of a element in a category with a G-action, and I too invoke the tao of mathematics (or possibly just a slightly more vigorous hand-wave ;)

If we consider the category of subgroups of a given 2-group (well, should probably be a 2-category, and we haven't fully got a handle on sub2-groups yet) with inclusions the morphisms (and I suppose transformations of inclusions too, to be safe) then I would imagine the above construction describes the kernel as its own stabilizer. Left and right mulitplication by an object of the 2-group are eqivalences by definition, and so isomorphisms

- x g: K --> Kg g x -: K --> gK

in the (2-)cat of subgroups of G, and a:g1 --> g2 will give us a (natural) isomorphism between

- x g1 and - x g2,

and

g1 x - and g2 x -.

Can we take this further and say that all sub2-groups of a 2-group G are the stabilizers of the subcategories of G, the action given by mulitplication? Many stabilizers will be trivial (or equivalent to trivial) - those of subcategories which aren't sub2-groups. This certainly works for the lattice of subsets of a honest group, and feels more Kleinian.

-----------

The other polished point is the free 2-group. Take a one-object 2-globular diagram (so a lot of edges from a vertex to itself with unoriented 2-cells thrown in). Take the free groupoid, or I suppose the fundamental bigroupoid, of it. I'm not sure if the interchange law comes for free or not.

We can then throw in some more isomorphisms = relations, possibly using a double 2-category construction, but one which is still a legit 2-category (this is just a bit easier for me to visualise). We get some empty tin cans and some full ones, and this should be done in a way such that the homotopy groups...

I just had this crazy realization: when I consider the homotopy groups of this free 2-group as above, what I'm doing is taking a 2D connected cell complex and forming its homology groups. This is why I love maths!

For a finite complex, the homotopy groups will be free groups modulo the relation `are homologous'. This also connects in a very nice way with the Betti numbers so far used.

Back when David was evil, he was worried that equivalent vector 2-spaces could have very different vector sub-2-spaces. For example,

1: C -> C

and

1: {0} -> {0}

are equivalent, but the former has a nontrivial sub-2-space, namely

i:{0} -> C

which is not a sub-2-space of the latter.

But now he's using a definition of "sub-2-group" where we say H is a sub-2-group of G if it's equipped with a faithful functor

i: H -> G

Now, according to the current definition, as i:{0} -> C is clearly faithful into C -> C, it must be a sub-2-space. So it had better also be a sub-2-space of {0} -> {0}. And, Lo and Behold, it is!

Of course this is just a consequence of using a non-evil definition of "sub-2-space". Non-evil means "invariant under equivalence", so of course any non-evil definition gives answers that are invariant under equivalence.

But this does not mean we have the correct definition of sub-2-space.

Indeed, you're jumping the gun slightly. I hadn't quite gotten around to officially proclaiming a definition of sub-2-group. In my Starbucks session I observed some properties of "stabilizer sub-2-groups" and guessed that these should be part of the definition of "sub-2-group". I also noticed some properties that shouldn't be part of the definition. But, I never claimed to have discovered all the properties of stabilizer sub-2-groups.

So, I still have a chance of weaseling out of any paradoxes you throw my way: I can claim there's some extra clause in the definition of sub-2-group that saves the day.

Or, if that doesn't save me, I can pull another trick.

Note that we're talking about sub-2-groups for two different reasons here. First, so we can talk about stabilizer sub-2-groups, which are fundamental to Klein 2-geometry. Second, so we can talk about vector sub-2-spaces, which are fundamental to projective 2-geometry.

It's great that we've got these two different reasons for thinking about sub-2-groups: it lets us study them from different angles. But, we can't be 100% sure that the same class of sub-2-groups will be the right thing for both purposes. Sometimes notions "split in two" when you generalize them. When you categorify, you always have to keep in mind this possibility. So, maybe this is what's going on.

(Of course I don't want to play that card prematurely - it's nicest when concepts categorify in a unified way, without fragmenting.)

Anyway, you're doing a great job of putting me on the hotseat, making me struggle to come up with a beautiful theory that's not "too weird" in one way or another. But, I don't feel you've got me boxed in yet.

You know, before I go to bed I just want to emphasize that this "splitting of concepts as we categorify" is already famous - precisely in the realm we're struggling with now. The twin concepts of "injective" and "surjective" for functions break apart into three famous concepts for functors: "faithful", "full" and "essentially surjective". The usual factorization of a function into a surjection followed by an injection breaks apart into a three-way factorization, as explained in section 3 here. And, this turns out to be not a mess, but part of a beautiful pattern.

So, it's quite possible that slugging it out over the puzzle "what's the correct generalization of `injection' for 2-groups?" is missing the point. We see two important concepts staring us in the face:

injective on pi_1 - "essentially injective"

injective on pi_2 - "faithful"

And, it may be necessary to deploy both of these, either separately or in combination, in the right ways at the right times.

The delights of dialectic. I've always wanted to live in a Lakatos dialogue. Actually, I think we're being far more respectful to each other than the characters in Proofs and Refutations.

injective on pi_1 - "essentially injective"

injective on pi_2 - "faithful"

And, it may be necessary to deploy both of these, either separately or in combination, in the right ways at the right times.

If you want to stick with your claim that 2-vector spaces can be classified by 2 Betti numbers, (b_0,b_1), and you insist on not being evil, then our current tools to decide about a potential subspace (c_0,c_1) offers us:

(i) faithful requiring that c_1 <= b_1

(ii) essentially injective on objects requiring that c_0 <= b_0

(iii) full on isomorphisms requiring that c_1 >= b_1

I'll bet (i) always gets picked. Tediousness is hardly the right way to judge things, but I'd rather not have (iii) as with (i) we'd have to have c_1 = b_1.

(ii) would mean that i:{0} -> C is not a subspace of 1:C -> C, or that the 2-group with two objects and only identity arrows is not a sub-2-group of the 2-group with two objects and single arrows between each pair.

I've seen a subobject in a category defined as an equivalence class of monics. Modulo the correct definition of injective for the case of 2-groups, would it be more correct to say a sub-2-group H of G is an arrow H >--> G, or an equivalence class of arrows?